PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 1 9JANUARY2004

The First Law for Slowly Evolving Horizons

Ivan Booth* Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland and Labrador, Canada A1C 5S7

Stephen Fairhurst† Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 (Received 5 August 2003; published 6 January 2004) We study the mechanics of Hayward’s trapping horizons, taking isolated horizons as equilibrium states. Zeroth and second laws of dynamic horizon mechanics come from the isolated and trapping horizon formalisms, respectively. We derive a dynamical first law by introducing a new perturbative formulation for dynamic horizons in which ‘‘slowly evolving’’ trapping horizons may be viewed as perturbatively nonisolated.

DOI: 10.1103/PhysRevLett.92.011102 PACS numbers: 04.70.Bw, 04.25.Dm, 04.25.Nx

The laws of black hole mechanics are one of the most Hv, with future-directed null normals ‘ and n. The ex- remarkable results to emerge from classical general rela- pansion ‘ of the null normal ‘ vanishes. Further, both tivity. Recently, they have been generalized to locally the expansion n of n and L n‘ are negative. defined isolated horizons [1]. However, since no matter This definition captures the local conditions by which or radiation can cross an isolated horizon, the first and we would hope to distinguish a black hole. Specifically, second laws cannot be treated in full generality. Instead, the expansion of the outgoing light rays is zero on the the first law arises as a relation on the phase space of horizon, positive outside, and negative inside. Addition- isolated horizons rather than as a truly dynamical rela- ally, the ingoing null rays are converging. As we will see, tionship. Even existing physical process versions of the these horizons can be spacelike or null and include both first law [2,3] consider transitions between infinitesimally equilibrium and nonequilibrium states—an important separated isolated (or Killing) horizons. In this Letter, we feature for our perturbative study. By comparison, if introduce a framework which allows us to extend the first one is interested only in the dynamical phase, the space- law to all slowly evolving horizons, even those whose like dynamical horizons recently introduced by Ashtekar initial and final states are not infinitesimally separated. and Krishnan [5] are more relevant. However, such hori- Additionally, we provide a simple characterization of zons are always expanding and thus not so suitable for how close a horizon is to equilibrium, which will be studying transitions to and from equilibrium. useful in the final stages of numerical simulations of Hayward [4] has extensively studied the properties of black hole collisions. To obtain such a law, we first need FOTHs. Here, we summarize only those of his results a local dynamical definition of a black hole horizon. which are important for our work. First, consider the Hayward has introduced future outer trapping horizons quantities associated with the null vector fields ‘ and n, (FOTHs) for exactly this purpose. Furthermore, he has which we normalize so that ‘ n 1. We denote their shown that they necessarily satisfy the second law—their relative expansions ‘ and n. Their twists are zero area cannot decrease in time [4]. since they are normal to the Hv cross sections of the ‘ n In this Letter, we will show that both the zeroth and first horizon. Finally, we write their shears as ab and ab . laws are also applicable to FOTHs. In order to do so, we Next, for each choice of the fields ‘ and n, there exists a introduce dynamical notions of surface gravity and an- scalar field C on H so that gular momentum which are applicable to all such hori- a ‘a Cna and  ‘ Cn (1) zons. It follows immediately that the surface gravity is V a a a necessarily constant if the horizon is in equilibrium (i.e., are, respectively, tangent and normal to the horizon. Note isolated). Next, we introduce the notion of a slowly evolv- that V V   2C. Hayward [4] has shown that, ing horizon, for which the gravitational and matter fields if the null energy condition holds, then are slowly changing. It is only in this limited context that we expect to obtain the dynamical first law.We will show C 0 (2) that this is indeed the case. on a FOTH. Thus, the horizon must be either spacelike or Let us begin by recalling Hayward’s definition. null, and the second law of trapping horizon mechanics Future outer trapping horizon.—A future outer trap- follows quitep easily. If q~ab is the two metric on the cross ping horizon (FOTH) is a smooth three-dimensional sections, and q~ is the corresponding area element, then submanifold H of space-time which is foliated by a p p preferred family of spacelike two-sphere cross sections L V q~ Cn q~: (3)

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By definition, n is negative and we have just seen that C which can be written as is non-negative. Hence, we obtain the local second law:p If b the null energy condition holds, then the area element q~ !a :nbr ‘ ; (4) of a FOTH is nondecreasing along the horizon [4]. a Clearly, on integration, the same law applies to the total where the arrow signifies that the derivative is pulled back area of the horizon. It is nondecreasing and remains to the horizon. Then, the isolated horizon surface gravity constant if and only if the horizon is null. is given by  ‘a! while angular momentum infor- To further restrict the rescaling freedom of the null ‘ a mation is contained in the other components of ! . vectors, we require that v 1, where v is the folia- a L V Now, ! written in this form [Eq. (4)] continues to be tion parameter labeling the cross sections. Thus, choosing a an intrinsically well-defined quantity for trapping hori- the foliation parameter fixes the length of a, a, and the V zons. Then, an obvious generalization is to define null vectors. However, at this stage, we are still free to a change the foliation labeling and, by doing so, rescale V a a b v : V !a nbV raV and and the null vectors by functions of v. From the isolated Z 1 p (5) horizon perspective, the normalization we have chosen is : 2 a a a J’ d x q~’ !~a; very natural, as in that case V ‘ , and it is customary 8 Hv to choose the foliation parameter v, so that n dv. Physical characterization of trapping horizons.—The where ’a is any vector field tangent to the horizon cross laws of black hole mechanics are given in terms of sections, and !~a is the projection of !a into the two- a physical quantities such as energy, angular momentum, surface Hv. In the case where ’ is divergence free, J’ is surface gravity, and area. One of the advances of the independent of the normalization of ‘ and equal to other isolated horizon formalism was to provide definitions of popular expressions for angular momentum (such as the all these quantities at the horizon, without reference to Komar, Brown-York [6], or Ashtekar-Krishnan [5] defi- spacelike infinity or the space-time in which the horizon nitions). It is clear that both the surface gravity v and is embedded [1,2]. In generalizing these definitions to angular momentum J’ expressions reduce to their iso- FOTHs, we will be motivated by two requirements: (i) the lated horizon values if the horizon is null. new definitions should match the old ones when trapping The constraint law.—Consider the set of vector fields in horizons are isolated and (ii) the new definitions should H that generate one-parameter families of diffeomor- depend only on quantities that are intrinsic to the horizon, phisms that map two-dimensional cross sections Hv of and as such truly be properties of the horizon itself. That the horizon into each other. Any such vector field may be a a a said, the ultimate justification for these expressions will written in the form X x0V x~ , for some function a be found in their utility in the calculations that follow. x0v and vector field x~ that is everywhere tangent to an For isolated horizons, both surface gravity and angular appropriate cross section. Then, by integrating the aXb momentum were defined with respect to a one-form !a component of the Einstein equations over Hv, we obtain the following relationship: Z p p Z p Z p 1 2 a 2 a b 1 2 ‘ab nab d x‰vLX q~ x~ LV q~!~aŠ d x q~‰TabX  Š d x q~ C LXq~ab 8G Hv Hv 16G Hv Z p p 1 2 d x‰2C q~LXnCnLX q~Š: (6) 16G Hv A full derivation of this result will be given in [7], and in another paper we will see that this relation plays a crucial dE TdS work terms: (7) role in the Hamiltonian formulation of general relativity on manifolds with FOTHs as boundaries [8]. Here, we Furthermore, in the general case, there is no clear inter- show that in certain restricted situations Eq. (6) becomes pretation of the right-hand side of (6) as a flux of energy a first law for dynamical black holes. through the horizon. Instead, we will require the horizon Quasistationary horizons.—At first glance, one might to be ‘‘quasistationary’’ and then obtain a first law. think that (6) is already the dynamical first law of black Heuristically, it is clear that the properties of a quasi- hole mechanics. After all, it relates rates of change of area stationary horizon, such as the area and surface gravity, and angular momentum to fluxes across the horizon. should be slowly varying. However, since there is a re- However, there are several reasons why this is not so. scalingp freedomp in the vector field V , requiring The first and most important is that we should not expect LV q~= q~ to be small is not a meaningful condi- the standard a_ form of the first law to hold in all tion—it can be satisfied on any trapping horizon by dynamical situations. In thermodynamics, it is only in suitably rescaling V . Furthermore, we cannot fix the the quasistatic case that it is possible to write the energy norm of V (or its average) to unity since we are interested balance equation as in the limit as V becomes null. Instead, we introduce the 011102-2 011102-2 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 1 9JANUARY2004

: one-formp ap dav satisfying V 1, and require gravity is a slowly varying function and we can write LV q~L q~=q~ to be small compared to the charac- 0 1 teristic scale of the horizon. This condition is invariant v   ; (10) under rescalings, and the expression vanishes whenever where 0 is a constant. Recall that an isolated horizon thep horizon p is null. On a spacelike horizon, 2 corresponds to  0. Hence, we immediately obtain the LV q~L q~qqC~ n and so is closely related to the expansion of the surface [see Eq. (3)]. By evaluating zeroth law: The surface gravity of a FOTH is constant if this expression, we can invariantly determine the expan- the horizon is isolated. sion rate of the horizon and choose the foliation accord- The dynamical first law.—We can now derive the first law for slowly evolving horizons. First, we consider evo- ingly. Then, we require other fields to be slowly varying a with respect to this foliation. lution with respect to V , an appropriate evolution vector for a horizon with no obvious axis of symmetry—a Slowly evolving horizon.—A region of a future outer a nonrotating horizon. Setting x~ 0 and x0 1 in the trapping horizon H with v 2‰v1;v2Š is slowly evolving (at a rate ) if there exists a parameter   1 such that (i) constraint law (6) and expanding in powers of ,we on every cross section H with v 2‰v ;v Š, find that all terms vanish at zeroth and first order. Then, v 1 2 to 2, the first law for a slowly evolving horizon reads Z p O d2x qqabr qcdr 2: ~ ~ aV b ~ c d 1 0 _ Hv  a_ H E 8G Z p  (ii) The foliation parameter v is chosen so that jV j 2 a b 1 ‘ 2 p : d x q~ Tab‘ ‘ j j ; 2C , and its rates of change are similarly small over Hv 8G the horizon. (iii) The one-form !~a and expansion n are (11) 2 2 slowly evolving: jLV !~aj=rH and jLV nj=rH, 2 where the dot signifies a derivative with respect to v.It where rH is the area radius of the horizon, aH 4rH. ~ 2 n 2 a b 2 says that the surface gravity multiplied by the change in (iv) jRj, j!j , j j ,andTabn n 1=rH or smaller. The first condition gives an invariant characterization area is equal to a flux of energy through the horizon. This of slow expansion. The second condition restricts the flux is comprised of two terms, both of which are positive. foliation to guarantee that the area is slowly evolving The first is the flux of matter through the horizon and is a 2 familiar from standard physical process versions of the with respect to V , specifically LV rH . The third requires other geometric quantities to also be slowly first law [3]. The second term is a flux of gravitational evolving, while the fourth fixes a reasonable horizon shear through the horizon—which would naturally be geometry and demands that conditions in the surrounding interpreted as a flux of gravitational radiation. A similar term has been obtained previously by Hawking [9] when space-time not be too extreme; here R~ is the Ricci scalar considering perturbations of an event horizon. of the two surfaces Hv. An immediate consequence of our definition is that isolated horizons are examples of slowly We would like to ‘‘integrate’’ (11) in order to obtain an expression for the energy of the horizon. To this end, we evolving horizons with  C 0 [provided condition a (iv) is satisfied]. further restrict the choice of V by requiring that

Let us now consider the implications of this definition. 0 1 Here, we will simply state the consequences; a more  : (12) 2rH complete discussion will be given in [7]. For concreteness, we restrict the allowed matter to be scalar and/or electro- Note that while we cannot prove that this rescaling is magnetic fields. From a projection of the Einstein equa- possible on every slowly evolving horizon, we can show tions and condition 2, we can show that that, if the horizon satisfies a certain genericity condition, it will be. In particular, perturbations to Schwarzschild ‘ a b 2 jab j and Tab‘ ‘  : (8) and nonextremal Kerr horizons will satisfy the condition. 0 Since  is a function of rH alone, we can integrate the (Here and in the following, we choose to focus on the  a_ term in (11) to obtain the usual expression for the factors. The powers of rH required to make the equations energy dimensionally correct are omitted.) Then, the allowed a b rH form of the matter fields forces Tab‘ x~  and E : (13) a b 2G LV Tab‘ n . Further application of the Einstein equations and Bianchi identities gives the Weyl compo- Rotating horizons.—Next, consider rotating horizons. ~ nents 0; 1  and L V R . On an isolated hori- In (5), we gave a definition for the angular momentum. zon, each of these quantities would be zero. Lastly, However, this quantity only has a physical significance if condition (iii) is sufficient to guarantee that the vector field ’a is an (approximate) symmetry of the horizon. Hence, we define the following. jL  j and jq~ br  j: (9) V v a b v Approximately symmetric horizon.—The vector field Therefore, on a slowly evolving horizon, the surface ’a is an approximate symmetry of a trapping horizon if 011102-3 011102-3 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 1 9JANUARY2004

(i) ’a is tangent to the cross sections of the horizon, This E is equal to the energy of a rotating isolated horizon generates a family of closed integral curves, and is nor- with the preferred choice of normalization for ‘, and, in malized so that those curves have affine parameter length particular, is the mass of a Kerr black hole with parame- a a 2.(ii)’ is Lie dragged up the horizon by V ,i.e., ters aH and J’. a a L V ’ 0.(iii)’ is an approximate symmetry of the The notion of a slowly evolving horizon provides a ab 2 horizon geometry and matter fields: jq~ ra’bj < , bridge between the equilibrium of isolated horizons and a b jL’q~abj <, jL’!~aj <, jL’nj <,andjTab’ ‘ j < the fully dynamical horizons of [5]. In particular, our first 2. (These conditions are analogous to those satisfied by law (17) bears a striking resemblance to the dynamical V a on a slowly evolving horizon.) horizon energy balance formula. There are, however, For a slowly evolving horizon which is approximately several important differences. First, we restrict to slowly symmetric, the angular momentum J’ is a meaningful evolving horizons, which are near to equilibrium. Doing quantity. Going back to the constraint law and setting so enables us to provide a locally defined surface gravity a a x0 0 and x~ ’ , we find its time rate of change. As in v which is shown to be slowly varying over the horizon. the nonrotating case, Eq. (6) will vanish at zeroth and first This surface gravity maintains its usual interpretation as orders in . To second order, we have the acceleration of a vector field, V a, along the horizon. Z   j j2 p 1 Second, we do not obtain an equivalent of the term of L J d2x q~ T a’b ‘abL q~ : [5], which is likely related to angular momentum. V ’ ab 16G ’ ab Hv However, we expect that term would vanish at the order (14) of perturbation theory which we are considering. In the Thus, the rate of change of angular momentum of the future, we plan to examine the connection between these horizon depends upon the flux of matter and gravitational two formulations in more detail and determine in what fields through the horizon. precise sense our first law can be seen as a perturbative To get a general first law, we would like to combine the form of the dynamical horizon energy balance formula. rate of change of angular momentum with the first law for In summary, we have examined conditions for which a nonrotating horizons (11). To do this, we again fix the slowly evolving FOTH may be said to be ‘‘perturbatively average value of the surface gravity as well as an angular nonisolated.’’ This has allowed us to obtain a truly dy- velocity (we are following a similar strategy to that namical version of the first law of black hole mechanics. taken in [5], though with the caveat that here only the However, we expect that the notion of a slowly evolving average value of the surface gravity is fixed). We do this horizon will also find application in numerical studies of by requiring that both take the same values as in a Kerr how horizons settle down to equilibrium. Specifically, space-time with the same J and a : one could track the approach of the parameter  to zero. ’ H We thank A. Ashtekar, C. Beetle, and B. Krishnan for 2GJ r4 4G2J2 discussions. S. F. was supported by the Killam Trusts at : q’ and 0 qH ’ : the University of Alberta and NSF Grant No. PHY- r r4 4G2J2 2r3 r4 4G2J2 H H ’ H H ’ 0200852 at UWM; I. B. was supported by NSERC. (15) With area and angular momentum slowly changing up the horizon, the angular velocity is also slowly varying. Finally, we consider the full constraint law (6) with the a evolution vector field X chosen as *Electronic address: [email protected] † ta : V a ’a: (16) Electronic address: [email protected] [1] A. Ashtekar, C. Beetle, and J. Lewandowski, Phys. Rev. Then, we can once again expand out the constraint law D 64, 044016 (2001); A. Ashtekar et al., Phys. Rev. Lett. (6) to order 2 and this time obtain 85, 3564 (2000); A. Ashtekar, S. Fairhurst, and B. Krishnan, Phys. Rev. D 62, 104025 (2000). 1 E_ 0a_ J_ ; where [2] A. Ashtekar, C. Beetle, and S. Fairhurst, Classical 8G ’ 17 Z   (17) Quantum Gravity , 253 (2000). p 1 [3] S. Gao and R. M. Wald, Phys. Rev. D 64, 084020 (2001). _ 2 a b ‘ab E : d x q~ Tabt   Ltq~ab : [4] S. A. Hayward, Phys. Rev. D 49, 6467 (1994). H 16G v [5] A. Ashtekar and B. Krishnan, Phys. Rev. Lett. 89, 261101 0 Since  and are specific functions of only rH and J’, (2002); Phys. Rev. D 68, 104030 (2003). it is once again possible to integrate the left-hand side of [6] J. D. Brown and J.W. York, Phys. Rev. D 47, 1407 (1993). the equation to obtain as an expression for the energy [7] I. Booth and S. Fairhurst (to be published). q [8] I. Booth and S. Fairhurst (to be published). 4 2 2 [9] S. Hawking, The Event Horizon, in Black Holes,edited rH 4G J’ E : (18) by B. deWitt and C. deWitt (Gordon and Breach, New 2GrH York, 1973).

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